To any complex smooth variety Y with an action of a finite group G, Etingof associates a global Cherednik algebra. The usual rational Cherednik algebra corresponds to the case of Y= C^n and a finite Coxeter group G
The question of estimating the length of short conjugators in between
elements in a group could be described as an effective version of the
conjugacy problem. Given a finitely generated group $G$ with word metric
$d$, one can ask whether there is a function $f$ such that two elements
$u,v$ in $G$ are conjugate if and only if there exists a conjugator $g$ such
that $d(1,g) \leq f(d(1,u)+d(1,v))$. We investigate this problem in free
solvable groups, showing that f may be cubic. To do this we use the Magnus
embedding, which allows us to see a free solvable group as a subgroup of a
particular wreath product. This makes it helpful to understand conjugacy
length in wreath products as well as metric properties of the Magnus
embedding.
Consider a diagram of the unknot with c crossings. There is a
sequence of Reidemeister
moves taking this to the trivial diagram. But how many moves are required?
In my talk, I will give
an overview of my recent proof that there is there is an upper bound on the
number of moves, which
is a polynomial function of c.
We will discuss the origin of the conjectured colour-kinematics
duality in perturbative gauge theory, according to which there is a
symmetry between the colour dependence and the kinematic dependence of the
S-matrix. Based on this duality, there is a prescription to obtain gravity
amplitudes as the "double copy" of gauge theory amplitudes. We will first
consider tree-level amplitudes, where a diffeomorphism algebra underlies
the structure of MHV amplitudes, mirroring the colour algebra. We will
then draw on the progress at tree-level to consider one-loop amplitudes.
Topological field theories give a connection between
topology and algebra. This connection can be exploited in both
directions: using algebra to construct topological invariants, or
using topology to prove algebraic theorems. In this talk, I will
explain an interesting example of the latter phenomena. Radford's
theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a
finite tensor category the quadruple dual functor is easy to
understand. It's somewhat mysterious that the double dual is hard to
understand but the quadruple dual is easy. Using topological field
theory, we show that Radford's theorem is exactly the consequence of
the Dirac belt trick in topology. That is, the double dual
corresponds to the generator of $\pi_1(\mathrm{SO}(3))$ and so the
quadruple dual is trivial in an appropriate sense exactly because
$\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2$. This is part of a large
project, joint with Chris Douglas and Chris Schommer-Pries, to
understand local field theories with values in the 3-category of
tensor categories via the cobordism hypothesis.
I will talk about the fixed-point properties of matrix groups acting CAT(0) paces, spheres and acyclic manifolds. The matrix groups include general linear groups, sympletic groups, orthogonal groups and classical unitary groups over general rings. We will show that for lower dimensional CAT(0) spaces, the group action of a matrix group always has a global fixed point and that for lower dimensional spheres and acyclic manifolds, a group action by homeomorphisms is always trivial. These results give generalizations of results of Farb concerning Chevalley groups over commutative rings and those of Bridson-Vogtmann, Parwani and Zimmermann concerning the special linear groups SL_{n}(Z) and symplectic groups Sp_{2n}(Z).